|
%I #18 Feb 22 2023 09:58:36
%S 1,0,1,0,2,0,0,3,2,0,0,5,4,0,0,0,7,11,1,0,0,0,11,20,6,0,0,0,0,15,40,
%T 16,0,0,0,0,0,22,68,40,3,0,0,0,0,0,30,120,91,11,0,0,0,0,0,0,42,195,
%U 186,41,0,0,0,0,0,0,0,56,320,367,105,3,0,0,0,0,0,0
%N Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.
%H Andrew Howroyd, <a href="/A330463/b330463.txt">Table of n, a(n) for n = 0..1325</a> (rows n = 0..50)
%F G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - _Andrew Howroyd_, Dec 29 2019
%e Triangle begins:
%e 1
%e 0 1
%e 0 2 0
%e 0 3 2 0
%e 0 5 4 0 0
%e 0 7 11 1 0 0
%e 0 11 20 6 0 0 0
%e 0 15 40 16 0 0 0 0
%e 0 22 68 40 3 0 0 0 0
%e ...
%e Row n = 5 counts the following sets of multisets:
%e {{5}} {{1},{4}} {{1},{2},{1,1}}
%e {{1,4}} {{2},{3}}
%e {{2,3}} {{1},{1,3}}
%e {{1,1,3}} {{1},{2,2}}
%e {{1,2,2}} {{2},{1,2}}
%e {{1,1,1,2}} {{3},{1,1}}
%e {{1,1,1,1,1}} {{1},{1,1,2}}
%e {{1,1},{1,2}}
%e {{2},{1,1,1}}
%e {{1},{1,1,1,1}}
%e {{1,1},{1,1,1}}
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
%p combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p seq(T(n), n=0..14); # _Alois P. Heinz_, Dec 30 2019
%t ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
%t Table[Length[Select[ppl[n,2],And[UnsameQ@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
%t (* Second program: *)
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
%t PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
%t T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, May 18 2021, after _Alois P. Heinz_ *)
%o (PARI)
%o A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n],n))}
%o {my(T=A(12)); for(n=1, #T, print(T[n]))} \\ _Andrew Howroyd_, Dec 29 2019
%Y Row sums are A261049.
%Y Column k = 1 is A000041.
%Y Multisets of multisets are A061260, with row sums A001970.
%Y Sets of sets are A330462, with row sums A050342.
%Y Multisets of sets are A285229, with row sums A089259.
%Y Sets of disjoint sets are A330460, with row sums A294617.
%Y Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.
%K nonn,tabl
%O 0,5
%A _Gus Wiseman_, Dec 19 2019
|
